Round-six lemma `T0n_eq`

Proved

NLogN.KFL.T0n_eq

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.KFLMath):

theorem NLogN.KFL.T0n_eq {K : ℕ} (hK : 1 ≤ K) (ρ : ℕ) (x y : ℕ) :
    ∀ n, T0n K ρ n x y = ((y ^^^ (x &&& selM K ρ n) : ℕ) : ℤ)

Context. KFL §2: Lemma 13 on chunk triples (canonical layout x, y, z = slots 0, 1, 2)

selM K ρ n = Σ_{i<n} 2^{ρ+iK} (the selected-position mask), testBit_selM.
T0y_eq: the ideal map T0 of ButterflyPacked is y ↦ y ⊕ (x ∧ selM (f-1)); guardN_xor: it preserves every guard.
The pull maps of the eight lines as triple maps lineP (realised by rot2, after a slot swap for the y-lines), their composite FSc = lineP l₁ ∘ … ∘ lineP l₈ and the forward S0c; S0c_FSc, FSc_S0c (inverse bijections), FSc_good (off the guard set FSc = T0c).
The correction: keyc = T0c ∘ FSc; bad_keyc (bad to bad), keyc_S0c (onto the bad set), T0c_keyc (the corrected pull is T0c).
T0c_XLc: with the last selected bit (flip1) the composite is the row addition RAc.

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFL
open Butterfly

Formal statement

theorem NLogN.KFL.T0n_eq {K : ℕ} (hK : 1 ≤ K) (ρ : ℕ) (x y : ℕ) :
    ∀ n, T0n K ρ n x y = ((y ^^^ (x &&& selM K ρ n) : ℕ) : ℤ) := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `NLogN.KFL.T0n_eq`, Solutions/KFLMath.lean (lines 120–173) of the round-six development.

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